No one answered it in a while, so I tested that in Showdown! to prove it.
In this replay, at turn 1 Tyranitar's Pursuit was executed first, after which Dialga's Dragon Tail was done at the last. Dragon Tail and other moves like Roar and Whirlwind always have -6 priority at making the target switch out, so Pursuit doesn't hit such opponents at double damage.
Pursuit is a special exception to the general rule of priority, due to its effect. Switching fundamentally happens before any moves can be performed, but when Pursuit is targeting a Pokémon that switches out, it will hit the Pokémon before it can switch, meaning that it will go before any other move, no matter its priority.
Switching gets performed always before anything, and Pursuit hits such targets with 2x damage and executes earlier that any move regardless of the speed.
This does not apply if used on an ally switching out, on a fleeing Pokémon, on a Pokémon using Teleport or Baton Pass, or on a Pokémon being forced to switch out.
Circle Throw, Dragon Tail, Roar, and Whirlwind all force the Pokemon to switch out and are thus unaffected by Pursuit.
Hope this helps!